KinexiaLEARN PHYSICS

Trigonometry — Height of a Loading Ramp (Worked Example)

Easy
Problem

A 5.0 m loading ramp makes a 30° angle with the horizontal ground. (a) How high is the top of the ramp? (b) What horizontal distance does it cover?

Right triangle formed by a 5.0 m ramp at 30 degrees showing height 2.5 m and run 4.33 m30°rampNW=mgf
The ramp as a right triangle: height opposite the 30° angle.

Given

  • hypotenuse = 5.0 m

  • θ = 30° above horizontal

Find

Vertical height h and horizontal run x.

Principle

In a right triangle, the sides are the hypotenuse scaled by sine and cosine of the angle — SOH CAH TOA.

Formula

Solution

  1. Height (opposite): h=5.0×sin⁡30°=5.0×0.5h = 5.0 \times \sin30° = 5.0 \times 0.5

  2. Calculate: h=2.5 mh = 2.5\ \mathrm{m}

  3. Run (adjacent): x=5.0×cos⁡30°=5.0×0.8660x = 5.0 \times \cos30° = 5.0 \times 0.8660

  4. Calculate: x=4.33 mx = 4.33\ \mathrm{m}

h = 2.5 m; x ≈ 4.33 m

Unit check

m × unitless trig ratio = m. Both are lengths — consistent.

What this means physically

A 30° ramp climbs exactly half its length — sin30° = 0.5 is worth memorizing. The 4.33 m run tells you how much floor space the ramp needs, which is the practical number for loading docks.

Common mistake

Using sine for the adjacent side. Opposite/hypotenuse is sine; adjacent/hypotenuse is cosine — sketch the triangle and label opposite the angle first.